Theorem 2.2 [1]}

Let \(U,V\)  be open subsets of \(\mathbb {R}^d\) .
If \(f:U \rightarrow V\)  is an analytic map and \(\operatorname{rank}df(x) = d\)  for some \(x \in U\) ,
then there exists a neighbourhood \(U^{\prime } \subset U\)  of \(x\)  so that \(f|_{U^{\prime }}\)  is a local analytic diffeomorphism.
Furthermore,
if \(f\)  is injective and \(\operatorname{rank}df(x) = d\)  for all \(x \in U\) , \(f\)  is an analytic diffeomorphism.

